In probability and statistics, the log-logistic distribution (known as the Fisk distribution in economics) is a continuous probability distribution for a non-negative random variable. It is used in survival analysis as a parametric model for events whose rate increases initially and decreases later, as, for example, mortality rate from cancer following diagnosis or treatment. It has also been used in hydrology to model stream flow and precipitation, in economics as a simple model of the distribution of wealth or income, and in networking to model the transmission times of data considering both the network and the software. The log-logistic distribution is the probability distribution of a random variable whose logarithm has a logistic distribution.↵ It is similar in shape to the log-normal distribution but has heavier tails . Unlike the log-normal, its cumulative distribution function can be written in closed form.
Characterization There are several different parameterizations of the distribution in use. The one shown here gives reasonably interpretable parameters and a simple form for the cumulative distribution function [1]. The parameter α > 0 {\displaystyle \alpha >0} is a scale parameter and is also the median of the distribution. The parameter β > 0 {\displaystyle \beta >0} is a shape parameter. The distribution is unimodal when β > 1 {\displaystyle \beta >1} its dispersion decreases as it β {\displaystyle \beta } increases. The cumulative distribution function is
F ( x ; α , β ) = 1 1 + ( x / α ) − β = ( x / α ) β 1 + ( x / α ) β = x β α β + x β {\displaystyle {\begin{aligned}F(x;\alpha ,\beta )&={1 \over 1+(x/\alpha )^{-\beta }}\\[5pt]&={(x/\alpha )^{\beta } \over 1+(x/\alpha )^{\beta }}\\[5pt]&={x^{\beta } \over \alpha ^{\beta }+x^{\beta }}\end{aligned}}}
where x > 0 {\displaystyle x>0} , α > 0 {\displaystyle \alpha >0} , β > 0. {\displaystyle \beta >0.}
The probability density function is
f ( x ; α , β ) = ( β / α ) ( x / α ) β − 1 ( 1 + ( x / α ) β ) 2 {\displaystyle f(x;\alpha ,\beta )={\frac {(\beta /\alpha )(x/\alpha )^{\beta -1}}{\left(1+(x/\alpha )^{\beta }\right)^{2}}}}
Alternative parameterization An alternative parametrization is given by the pair μ , s {\displaystyle \mu ,s} in analogy with the logistic distribution:
μ = ln ( α ) {\displaystyle \mu =\ln(\alpha )}
s = 1 / β {\displaystyle s=1/\beta }
Properties
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